How do I make a rectangular mesh subdivided evenly with square faces / quads?
Use loop cuts to manually subdivide the first time.  CTRL-R

Start by making a certain number of loop cuts across the short side.  Just enough to know loop cuts on the long side will result in a square.



Then make the necessary loop cuts on the long side to complete the square (or close enough to it.  In this example, I only needed one more on the long side



From there, you can subdivide normally and keep your square proportions.  If you really want to get perfect about it, and the dimensions of your plane are conducive to it, you could calculate the number of loop cuts needed by subdividing the dimensions of your plane mathmatically.  But usually not necessary.